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Table of Contents
Chapter 1 – Review of Algebra…page 5
Chapter 2 – Word Problems… page
Chapter 3 – Factoring …………page
Chapter 4- Algebraic Fractions… page
Chapter 5- Graphing Equations…page
Chapter 6- Systems of Equations... Page
Chapter 7- Radicals ……………….Page
Chapter 8- Quadratic Equations…..Page
Table of Contents
Chapter 9 – Proofs
Chaoter 10- Transformations
Chapter 1- Algebra Review
Evaluating Expressions
Remember to put fractions in ( ) in calc
Remember to put any negative numbers
in ( ) in the calc… if there is an exponent
that goes after the close )
Chapter 1 Algebra Review
Order of Operations PEMDAS
(x +y)2 … in order to solve this re-write
twice and foil
If there is a negative in front of ( )
make sure to multiply all terms by a
negative
–ex. 40 –(3x+2) = 40 -3x -2
When multiplying polynomials with
exponents…add the exponents
When dividing polynomials with
exponents… subtract the exponents
Chapter 1 Algebra Review
Solving Equations
Multiply on either side of the
equation to get rid of the ( )…
DO NOT write ( ) again!
Combine like terms on each
side of the equation separately
Begin to move terms from one
side of the equation to the other
using inverse operations
Chapter 2- Word Problems
Number Word problems
5 less than twice a number… 2x-5
Product = *, Quotient = /,
Sum = +, Difference= -
Chapter 2 - Word Problems
Consecutive Integer Problems
Motion Problems
Coin Problems
Percent Problems
Investment
Age
Chapter 3 Factoring
LOOK FOR GCF!!!!!!!!!!
Look for a Difference of
2 Perfect Squares (a + b)
(a –b)
Look for a trinomial and
do factoring by
grouping!!!!
Factoring By Grouping
Multiply first number (not variable) by the
last number
Look for factors of this new number that
add to the center term
Re-write trinomial splitting the center term
into these two factors with the variable!!!
Draw (), factor out GCF from each group
If two binomials match, smile and write that
binomial once and the other numbers and
variables as the other binomial
71.(3y-2)(y-2) 87. (r-5s)(r+2s)
73.(2x+3)(x-1)89.(m+n)(5m-2n)
75.(3x+5)(x-1)
77.(5x-8)(x+1)
79.(2x-5)(2x-1)
81.(3x+4)(2x-1)
83.(x+5)(10x-1)
85.(x+2y)(x+y)
6. s(t-1)(t+1) 24. 4(5x+3y)(5x-3y)
8. 2(x-4)(x+4) 26. 3(x+1)(x+1)
10.2(3m-2)(3m+2) 28. x(x+5)(x+2)
12. 7(3c-1)(3c+1) 30. 2a(x-3)(x+2)
14.y(y-5)(y+5)
32. (z3-z)(z3+z)
16.a(2a+b)(2a-b) 34. (x2+2)(x+1)(x-1)
18.d(3b+1)(3b-1)36. (y+3)(y-3)(y-2)(y+2)
20 (x2+1)(x-1)(x+1)
22. 5(r+R)(r-R)
Chapter 4- Algebraic Fractions
Multiplication and Division of
EXPRESSIONSFactor all numerators and
denominators
For division: flip the second fraction
Cancel any like terms if one is in the
top and one is in the bottom
Multiply across (top times top and
bottom times bottom) and simplify
Adding and Subtracting
EXPRESSIONS
Find the lowest common
denominator…sometimes you have to
factor the bottoms to do this
Multiply each fraction by the missing
terms of the denominator
Add the tops together…make sure to
put in () and distribute if there is
subtraction
*DO NOT CANCEL BOTTOMS
Algebraic Equations
Follow the steps for adding and
subtracting… except after you create
like denominators you can cancel
them away
Finish solving the equation with only
the tops
Inequality Equations
Follow steps for regular equations
(with or without fractions)
If you are multiplying or dividing by a
negative number, you MUST flip the
inequality sign
Solving Equations with many
Variables
Always get all the terms that include
an (x) on one side of the equation and
all the other terms on the opposite
side
Usually you factor afterwards
Chapter 5- Graphing Equations
Y = mx + b … this is the form that all
Linear equations should be in if you are
going to graph them
M = slope = (y2-y1)/(x2-x1)= numerator is
the amount you rise, the denominator
is the amount you run (left)
B= y-intercept- point at which the line
crosses the y-axis
To graph: 1) plot b 2) then use slope for
rise and run
Graphing in Calc:
Press Y=
Make sure the Y= menu is clear
Type in the given equation in y=mx +b
form
Press graph to see the graph, if you
don’t see the graph, press zoom and
then hit # 6 zoom standard, then hit
graph again…now u should see the
graph
Press 2nd, then hit graph, this shows
the table for your given equation
Types of Slopes
Positive –rises from left to right
(m is pos)
Negative- falls from left to right
(m is neg)
Zero slope – horizontal line… y=?
Undefined slope or No slope – vertical
line… x = ?
Parallel lines have the same slope
Given two Points: Write the
equation of a line
Plug the two points into the slope
formula
Plug the slope into y = mx +b, for m
Plug in one of the two given points
into the equation with the slope, solve
for b
Then plug the slope and y-intercept
into y = mx+ b
Absolute Value Graphs
Y = |x|… this graph is a v that has its
center point on the origin
Y = |x| + a … this moves the v up or
down on the y-axis
Y = |x + a|… this moves the v left or
right… it moves in the opposite
direction of a
Y = a|x|…this makes the v wider or
thinner, if a is a fraction =wider, if a is a
number greater than 1 = thinner
Absolute Value Graphs Continued
Y = -|x|…v flips upside down
For the Calculator:
Press y=, press MATH, press right
arrow to NUM, press #1 abs,
put the part of the equation that is
inside the bars inside (), put all other
parts outside the ()… Y = 2|x-3|+4…
the x-3 goes inside the ()
x = |y|… sideways in the 1st and 4th
quadrant
Graphing Inequalities
Put the equation in y = mx + b form,
remember if you are dividing or multiplying
by a negative you must flip the inequality
sign
< or > then the line is dashed
≥ or ≤ then the line is solid
To see shading on the calc:
Press y=, move cursor to the far left so that
the line next to Y1 is blinking, then press
ENT either 3 or 4 times, 3 times for greater
than, 4 times for less than
Chapter 6- Systems of Equations
Consistent SystemInconsistent SystemDependent SystemThe 3 methods of solving a system
Graphically
Addition Method
Substitution Method
Word Problems
Chapter 7 – Radicals
How do I add or subtract radicals?
How do I multiply radicals?
Monomial*monomial
Binomial *binomial
How do I divide with radicals?
How do I solve Radical Equations?
How do I add/subtract with Radicals?
Simplify all radicals
List the factor sets for the number under the radical sign
and break the number into the factor set that has the
largest perfect square
Then take the square root of the perfect square and write it
on the outside of the radical sign
Variables- even exponents divide by 2 and write the
variable with that exponent on the outside of the radical
sign
Odd exponents- subtract one from the exponent and leave
this on the inside of the radical, then divide the even
exponent by 2 and write on the outside of the radical sign
ONLY ADD/SUB if numbers/variables under the radical are
exactly the same!!!
How do I multiply/divide radicals?
Multiply/divide the numbers/variables
outside the radical sign by other
numbers/variables outside the radical
sign and only multiply/divide interior
numbers/variables by interior
numbers/variables
Ex. (3√5)(4 √6) = 12 √30
Ex. 2 * √5= 2 √5
FOIL FOR BINOMIAL*BINOMIAL
Ex. (3√5 + 2)(4 √6 + 4)= 12 √30+12
√5+8√6+8
Rationalizing a Fraction
NEVER leave a radical in the
denominator of a fraction
Multiply the numerator and
denominator of the fraction by the
radical number… this will make the
radical go away in the denominator
because you are squaring it
Ex. (3√5)/(4 √6)…multiply
by√6/√6…answer is √30/8
Perfect Squares to 15 and 20
4,9,16,25,36,49,64,81,100,121,144,
169,196,225…400
Solving Radical Equations
Isolate the Radical portion of the
equation
Square both sides of the equation
If a binomial is being squared, make
sure to write twice and FOIL
Now the radical should have
disappeared… solve like a regular
equation…you should always check
your answers…you can get an
erroneous answer
Chapter 8- Quadratic Equations
How do I solve for the Roots of an equation?
(find the zeros of the equation)
Using the calculator:
Using Factoring:
Put equation in standard form so that it is equal to
zero
When taking the square root of both sides of an
equation you get a positive and negative answer
See chapter 3 for factoring steps
Using the Quadratic Formula:
X= -b +/- √b2 – 4ac
2a
Using the Calculator:
TO find the Roots:
Type equation into Y=
Press zoom, press zoom standard
Press 2nd, trace, #2 for Zero
Use cursor to go to the left side of the first
root, press ENT
Use cursor to go to the right side of the first
root, press ENT
Press ENT again…answer appears
Repeat for second root on right
Also you can use these steps to find a
minimum or maximum
Using the Calculator:
TO Graph a parabola with the vertex:
To find the vertex without the
calc…….x= -b/2a
Enter the equation in Y=, press graph
Press 2nd, Graph for Table
Look for symmetry in the y chart
The number in the center of the
symmetry is your vertex
Plot those 7 points on your graph!
Word Problems
Consecutive Integer, Even and Odd
X, x+1, x+2
Even/odd- x, x+2, x+4
Rectangle Problems:
Perimeter = 2L +2w
Area= L*W
Remember to always put a binomial
variable in () in an equation
Ex. The sum of the squares of two
consecutive integers…
X2 + (x+1)2
Proofs
Addition and Subtraction Proofs
Partition… a part + a part = whole
Substitution… the two pieces you are
trying to prove congruent
Use addition if you have 4 small pieces
or 3 small pieces (then you need
reflexive)
Use subtraction if you have 2 big pieces
and 2 small pieces or 2 big pieces and 1
small piece (then you need reflexive)
List of Theorems
Definition of Midpoint: a line has only one
midpt which cuts it in half
Definition of a Angle Bisector: it is a line
that cuts an ANGLE in half
Definition of a Line Bisector: it is a line that
goes to the MIDPT
Definition of a Median: a line drawn from a
vertex of a triangle to the midpt of a side
Definition of Altitude: a line drawn from a
vertex that is perpendicular to the side
(makes right angles)
List of Theorems
Perpendicular lines form right angles
All right angles are congruent
Vertical angles are congruent
Definition of Supplementary Angles
Supplements of congruent angles are
congruent
If two angles are congruent and
supplementary then they are right angles
Definition of Complementary angles
Complements of congruent angles are
congruent
List of Theorems
Isos. Triangle Theorem: if 2 base angles
are congruent, then 2 sides are congruent
In an isos. Triangle, if the line drawn from
the vertex angle is an altitude then it is also
the median and angle bisector
Definition of an Equilateral Triangle: all
sides and angles are congruent
Perpendicular Bisector Theorem: if given a
picture that looks like a kite with the 2 top
sides congruent and 2 bottom sides
congruent, then there is a perp. Bis.
Proving Triangles
SAS, ASA, SSS
CPCTC… use this after you prove 2
triangles congruent to prove other sides or
angles congruent
If you are given 2 sets of lines or angles
that are cut in half and you want to prove
that half of one line is congruent to half of
the other line then ….
Use Division, or halves of equal quantities
are equal or multiplication
Reflexive: a side or angle is congruent to
itself
All right angles are 90
All straight lines are 180 degrees
Chapter 10- Transformations
Line Reflection… r name of the line you reflect over
Point Reflection… Ro for origin or another point (x,y)
Rotation… Rdegrees
Dilation… Dnumber that you multiply the preimage by
Translation…T(x,y) that you add to the preimage
Glide Reflection… composition of translation and
reflection
Line Symmetry
Point Symmetry
Orientation, Direct Isometry and Opposite Isometry
Line and Point Reflections
r x-axis…(x,y) …(x, -y)
r y-axis…(x,y)…(-x,y)
r y=x …(x,y) …(y,x)
r y=-x…(x,y)…(-y,-x)
Ro…reflect over origin (x,y)…(-x,-y)
Count the boxes to the origin and
then go that distance from the origin
in the other direction
Rotations
R90=R-270…(x,y)…(-y,x)
R180=R-180…(x,y)…(-x,-y)
R270=R-90…(x,y)…(y,-x)
Count the distance that the preimage
is from the x-axis and make the image
this distance from the y-axis in the
correct quadrant. Count the distance
the preimage is from the y-axis and
make the image this distance from the
x-axis.
Dilation and Translation
Dilation – Da...(x,y)…(ax,ay)
Translation-T(a,b)…(x,y)…(x+a,y+b)
Glide Reflection… either T(x,y) ◦ r y=x
Or ry=x ◦ T(x,y)
◦….then… but you work from
RIGHT to LEFT
Defintions:
Orientation- the letters going around a
figure from right to left or left to right
Direct Isometry- is a transformation
that preserves distance and
orientation…translation, rotation,
point reflection
Opposite Isometry- preserves
distance but not orientation…line
reflection
Dilation – DOES NOT PRESERVE
DISTANCE
Definitions
Line Symmetry- you can cut it in half
in a direction or you can fold the
figure on top of itself perfectly in some
direction
Point Symmetry- turn your piece of
paper 180 degrees and the picture
should look the same!!!